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Properties

Label 2-24e2-16.11-c2-0-8
Degree $2$
Conductor $576$
Sign $0.472 - 0.881i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.01 + 6.01i)5-s − 8.23·7-s + (6.51 − 6.51i)11-s + (8.82 − 8.82i)13-s + 14.1·17-s + (23.7 + 23.7i)19-s − 9.42·23-s + 47.3i·25-s + (−23.7 + 23.7i)29-s + 24.4i·31-s + (−49.4 − 49.4i)35-s + (24.2 + 24.2i)37-s + 6.67i·41-s + (−0.897 + 0.897i)43-s + 25.2i·47-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)5-s − 1.17·7-s + (0.592 − 0.592i)11-s + (0.679 − 0.679i)13-s + 0.833·17-s + (1.24 + 1.24i)19-s − 0.409·23-s + 1.89i·25-s + (−0.820 + 0.820i)29-s + 0.787i·31-s + (−1.41 − 1.41i)35-s + (0.654 + 0.654i)37-s + 0.162i·41-s + (−0.0208 + 0.0208i)43-s + 0.537i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.472 - 0.881i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.061100741\)
\(L(\frac12)\) \(\approx\) \(2.061100741\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-6.01 - 6.01i)T + 25iT^{2} \)
7 \( 1 + 8.23T + 49T^{2} \)
11 \( 1 + (-6.51 + 6.51i)T - 121iT^{2} \)
13 \( 1 + (-8.82 + 8.82i)T - 169iT^{2} \)
17 \( 1 - 14.1T + 289T^{2} \)
19 \( 1 + (-23.7 - 23.7i)T + 361iT^{2} \)
23 \( 1 + 9.42T + 529T^{2} \)
29 \( 1 + (23.7 - 23.7i)T - 841iT^{2} \)
31 \( 1 - 24.4iT - 961T^{2} \)
37 \( 1 + (-24.2 - 24.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 6.67iT - 1.68e3T^{2} \)
43 \( 1 + (0.897 - 0.897i)T - 1.84e3iT^{2} \)
47 \( 1 - 25.2iT - 2.20e3T^{2} \)
53 \( 1 + (32.6 + 32.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (8.31 - 8.31i)T - 3.48e3iT^{2} \)
61 \( 1 + (68.4 - 68.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (-7.71 - 7.71i)T + 4.48e3iT^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + 52.8iT - 5.32e3T^{2} \)
79 \( 1 - 87.2iT - 6.24e3T^{2} \)
83 \( 1 + (-9.53 - 9.53i)T + 6.88e3iT^{2} \)
89 \( 1 + 146. iT - 7.92e3T^{2} \)
97 \( 1 - 101.T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43537826033533612750848088084, −9.892580509585561654423534159204, −9.236399481972516755602044516855, −7.913678101383353131225896208212, −6.84818461199999673204960949822, −6.05325584056530112495158628132, −5.60320205930497690532859796617, −3.42788513081103401990954873754, −3.11835742194415183132883083156, −1.41948078053818822252373277552, 0.872110188224412076756264663098, 2.14881716981579854923468968776, 3.65981664791925854209285675762, 4.84396809676839169597974699926, 5.84606695262347782713956472079, 6.50748844576551414244933157936, 7.69337084701489659737646792168, 9.079707239048819529208614325366, 9.430477594793638860147379596011, 9.922274472914617335787476077961

Graph of the $Z$-function along the critical line